Prerequisites & Notation

Before You Begin

This chapter develops the three canonical linear precoding strategies for the MU-MIMO downlink: MRT, ZF, and RZF/MMSE. We assume the reader is comfortable with the MIMO channel model, channel estimation, and the achievable rate framework from Chapters 1--5.

MIMO channel model and channel hardening(Review ch01)
Self-check: Can you write the downlink system model $\mathbf{y}_k = \mathbf{h}_k^H \sum_{j=1}^{K} \mathbf{v}_{j} s_j + w_k$ ?
Channel estimation: MMSE estimates and estimation error(Review ch03)
Self-check: Do you know how estimation error $\tilde{\mathbf{h}}_k = \mathbf{h}_k - \hat{\mathbf{h}}_k$ enters the rate expressions?
Achievable rate analysis with use-and-then-forget (UatF) bound(Review ch04)
Self-check: Can you write the UatF SINR expression and explain why the estimation error is treated as additional noise?
Matrix pseudo-inverse and Moore--Penrose inverse
Self-check: Can you compute $\mathbf{H}(\mathbf{H}^{H} \mathbf{H})^{-1}$ and explain when it exists?
MIMO broadcast channel capacity (ITA Ch. 16)
Self-check: Do you know that the BC capacity region is achieved by dirty-paper coding, not linear precoding?

Notation for This Chapter

Symbols introduced or heavily used in this chapter. See also the NGlobal Notation Table master table.

Symbol	Meaning	Introduced
$\mathbf{H} \in \mathbb{C}^{K \times N_t}$	Aggregate channel matrix (rows = user channels)	s01
$\mathbf{h}_k \in \mathbb{C}^{N_t}$	Channel vector of user $k$	s01
$\mathbf{v}_{k} \in \mathbb{C}^{N_t}$	Precoding vector for user $k$	s01
$\mathbf{W} \in \mathbb{C}^{N_t \times K}$	Precoding matrix $[\mathbf{v}_{1}, \ldots, \mathbf{v}_{\ntn{nusers}}]$	s02
$P_t$	Total transmit power budget	s01
$\sigma^2$	Noise variance at each user	s01
$\text{SINR}_k$	Signal-to-interference-plus-noise ratio for user $k$	s01
$R_k$	Achievable rate for user $k$ (bits/s/Hz)	s01
$\alpha$	Regularization parameter for RZF precoding	s03
$p_k$	Power allocation coefficient for user $k$	s01

← Ch 5 Maximum Ratio Transmission (MRT)